American Institute of Mathematical Sciences

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January  2015, 11(1): 199-216. doi: 10.3934/jimo.2015.11.199

Sensor deployment for pipeline leakage detection via optimal boundary control strategies

Chao Xu 1, , Yimeng Dong 1, , Zhigang Ren 1, , Huachen Jiang 2, and  Xin Yu 3,

1. 

State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou, Zhejiang 310027, China, China, China
2. 

Institute of Operations Research & Cybernetics, Zhejiang University, Hangzhou, Zhejiang 310027, China
3. 

Ningbo Institute of Technology, Zhejiang University, Hangzhou, Zhejiang 310027, China

Received  December 2012 Revised  January 2014 Published  May 2014

We consider a multi-agent control problem using PDE techniques for a novel sensing problem arising in the leakage detection and localization of offshore pipelines. A continuous protocol is proposed using parabolic PDEs and then a boundary control law is designed using the maximum principle. Both analytical and numerical solutions of the optimality conditions are studied.
Keywords: Offshore pipeline network, leakage detection, optimal control., continuation approach, multi-agent system, partial differential equations, Lagrangian sensor.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Chao Xu, Yimeng Dong, Zhigang Ren, Huachen Jiang, Xin Yu. Sensor deployment for pipeline leakage detection via optimal boundary control strategies. Journal of Industrial & Management Optimization, 2015, 11 (1) : 199-216. doi: 10.3934/jimo.2015.11.199
References:
[1]

N. Ahmed and K. Teo, Optimal Control of Distributed Parameter Systems,, North Holland, (1981).   Google Scholar

[2]

S. Anita, V. Arnautu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics,, Modeling and Simulation in Science, (2011).  doi: 10.1007/978-0-8176-8098-5.  Google Scholar

[3]

V. Arnautu and P. Neittaanmaki, Optimal Control from Theory to Computer Programs,, Kluwer Academic, (2003).  doi: 10.1007/978-94-017-2488-3.  Google Scholar

[4]

P. Barooah, P. Mehta and J. Hespanha, Mistuning-based control design to improve closed-loop stability margin of vehicular platoons,, IEEE Transactions on Automatic Control, 54 (2009), 2100.  doi: 10.1109/TAC.2009.2026934.  Google Scholar

[5]

S. Blazic, D. Matko and G. Geiger, Simple model of a multi-batch driven pipeline,, Mathematics and Computers in Simulation, 64 (2004), 617.  doi: 10.1016/j.matcom.2003.11.013.  Google Scholar

[6]

F. Bullo, J. Cortes and S. Martinez, Distributed Control of Robotic Networks (In Applied Mathematics Series),, Princeton University Press, (2009).   Google Scholar

[7]

M. Chen and D. Georges, Nonlinear optimal control of an open-channel hydraulic system based on an infinite-dimensional model,, in Proceeding of the Conference on Decision and Control, (1999).   Google Scholar

[8]

H. Cho and G. Hwang, Optimal design for dynamic spectrum access in cognitive radio networks under rayleigh fading,, Journal of Industrial and Management Optimization, 8 (2012), 821.  doi: 10.3934/jimo.2012.8.821.  Google Scholar

[9]

E. Chow, L. Hendrix, M. Herberg, S. Itoh, B. Kong, M. Lall and P. Srevens, Pipeline Politics in Asia: The Intersection of Demand, Energy Markets, and Supply Routes,, National Bureau of Asian Research, (2010).   Google Scholar

[10]

Y. Ding and S. Wang, Optimal control of open-channel flow using adjoint sensitivity analysis,, Journal of Hydraulic Engineering-ASCE, 132 (2006), 1215.  doi: 10.1061/(ASCE)0733-9429(2006)132:11(1215).  Google Scholar

[11]

Z. Feng, K. Teo and V. Rehbock, Branch and bound method for sensor scheduling in discrete time,, Journal of Industrial and Management Optimization, 1 (2005), 499.  doi: 10.3934/jimo.2005.1.499.  Google Scholar

[12]

Z. Feng, K. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time,, Automatica, 44 (2008), 1295.  doi: 10.1016/j.automatica.2007.09.024.  Google Scholar

[13]

G. Ferrari-Trecate, A. Buffa and M. Gati, Analysis of coordination in multi-agent systems through partial difference equations,, IEEE Transactions on Automatic Control, 51 (2006), 1058.  doi: 10.1109/TAC.2006.876805.  Google Scholar

[14]

P. Frihauf and M. Krstic, Leader-enabled deployment onto planar curves: A pde-based approach,, IEEE Transactions on Automatic Control, 56 (2011), 1791.  doi: 10.1109/TAC.2010.2092210.  Google Scholar

[15]

R. Glowinski, J. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach,, (Encyclopedia of Mathematics and its Applications) Cambridge University Press, (2008).  doi: 10.1017/CBO9780511721595.  Google Scholar

[16]

H. Hao and P. Barooah, On achieving size-independent stability margin of vehicular lattice formations with distributed control,, IEEE Transactions on Automatic Control, 57 (2012), 2688.  doi: 10.1109/TAC.2012.2191179.  Google Scholar

[17]

H. Hao, P. Barooah and P. Mehta, Stability margin scaling laws for distributed formation control as a function of network structure,, IEEE Transactions on Automatic Control, 56 (2011), 923.  doi: 10.1109/TAC.2010.2103416.  Google Scholar

[18]

J. Kim, K. Kim, V. Natarajan, S. Kelly and J. Bentsman, PdE-based model reference adaptive control of uncertain heterogeneous multiagent networks,, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1152.  doi: 10.1016/j.nahs.2008.09.008.  Google Scholar

[19]

J. Kim, V. Natarajan, S. Kelly and J. Bentsman, Disturbance rejection in robust PdE-based MRAC laws for uncertain heterogeneous multiagent networks under boundary reference,, Nonlinear Analysis: Hybrid Systems, 4 (2010), 484.  doi: 10.1016/j.nahs.2009.11.005.  Google Scholar

[20]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs,, SIAM, (2008).  doi: 10.1137/1.9780898718607.  Google Scholar

[21]

Z. Lin, Distributed Control and Analysis of Coupled Cell Systems,, VDM Verlag, (2008).   Google Scholar

[22]

W. Litvinov, Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions,, Journal of Industrial and Management Optimization, 7 (2011), 291.  doi: 10.3934/jimo.2011.7.291.  Google Scholar

[23]

M. Liu, S. Zang and D. Zhou, Fast leak detection and location of gas pipelines based on an adaptive particle filter,, International Journal of Applied Mathematics and Computer Science, 15 ().   Google Scholar

[24]

M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks (In Applied Mathematics Series),, Princeton University Press, (2010).   Google Scholar

[25]

T. Meurer and M. Krstic, Finite-time multi-agent deployment: A nonlinear pde motion planning approach,, Automatica, 47 (2011), 2534.  doi: 10.1016/j.automatica.2011.08.045.  Google Scholar

[26]

S. Moura and H. Fathy, Optimal boundary control & estimation of diffusion-reaction PDEs,, in Proceeding of the Conference on Decision and Control, (2011), 921.   Google Scholar

[27]

R. Murray, Recent research in cooperative control of multi-vehicle systems,, Journal of Dynamical Systems, (): 571.   Google Scholar

[28]

R. Olfati-Saber and R. Murray, Consensus problems in networks of agents with switching topology and time-delays,, IEEE Transactions on Automatic Control, 49 (2004), 1520.  doi: 10.1109/TAC.2004.834113.  Google Scholar

[29]

P. Parfomak, Pipeline Safety and Security: Federal Programs,, Congress Research Services (CRS) Report for Congress, (2008).   Google Scholar

[30]

M. Rafiee, Q. Wu and A. Bayen, Kalman filter based estimation of flow states in open channels using Lagrangian sensing,, Proceedings of the Conference on Decision and Control, (2009), 8266.  doi: 10.1109/CDC.2009.5400661.  Google Scholar

[31]

W. Ren and Y. Cao, Distributed Coordination of Multi-agent Networks,, (Communications and Control Engineering Series) Springer-Verlag, (2011).   Google Scholar

[32]

A. Sarlette and R. Sepulchre, A PDE viewpoint on basic properties of coordination algorithms with symmetries,, in Proceedings of the Conference on Decision and Control, (2009), 5139.  doi: 10.1109/CDC.2009.5400570.  Google Scholar

[33]

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Edition,, SIAM, (2004).  doi: 10.1137/1.9780898717938.  Google Scholar

[34]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications (Graduate Studies in Mathematics),, American Mathematical Society, (2010).   Google Scholar

[35]

G. Wang and H. Ye, Leakage Detection and Localization of Long Distance Fluid Pipelines,, Tsinghua University Press, (2010).   Google Scholar

[36]

Z. Wang, H. Zhang, J. Feng and S. Lun, Present situation and prospect on leak detection and localization techniques for long distance fluid transport pipeline,, Control and Instruments in Chemical Industry, 30 (2003), 5.   Google Scholar

[37]

S. Woon, V. Rehbock and R. Loxton, Global optimization method for continuous-time sensor scheduling,, Nonlinear Dynamics and Systems Theory, 10 (2010), 175.   Google Scholar

[38]

S. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems,, Optimal Control Applications and Methods, 33 (2012), 576.  doi: 10.1002/oca.1015.  Google Scholar

[39]

K. Yiu, K. Mak and K. Teo, Airfoil design via optimal control theory,, Journal of Industrial and Management Optimization, 1 (2005), 133.  doi: 10.3934/jimo.2005.1.133.  Google Scholar

[40]

C. Yu, B. Li, R. Loxton and K. Teo, Optimal discrete-valued control computation,, Journal of Global Optimization, 56 (2013), 503.  doi: 10.1007/s10898-012-9858-7.  Google Scholar

show all references

References:
[1]

N. Ahmed and K. Teo, Optimal Control of Distributed Parameter Systems,, North Holland, (1981).   Google Scholar

[2]

S. Anita, V. Arnautu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics,, Modeling and Simulation in Science, (2011).  doi: 10.1007/978-0-8176-8098-5.  Google Scholar

[3]

V. Arnautu and P. Neittaanmaki, Optimal Control from Theory to Computer Programs,, Kluwer Academic, (2003).  doi: 10.1007/978-94-017-2488-3.  Google Scholar

[4]

P. Barooah, P. Mehta and J. Hespanha, Mistuning-based control design to improve closed-loop stability margin of vehicular platoons,, IEEE Transactions on Automatic Control, 54 (2009), 2100.  doi: 10.1109/TAC.2009.2026934.  Google Scholar

[5]

S. Blazic, D. Matko and G. Geiger, Simple model of a multi-batch driven pipeline,, Mathematics and Computers in Simulation, 64 (2004), 617.  doi: 10.1016/j.matcom.2003.11.013.  Google Scholar

[6]

F. Bullo, J. Cortes and S. Martinez, Distributed Control of Robotic Networks (In Applied Mathematics Series),, Princeton University Press, (2009).   Google Scholar

[7]

M. Chen and D. Georges, Nonlinear optimal control of an open-channel hydraulic system based on an infinite-dimensional model,, in Proceeding of the Conference on Decision and Control, (1999).   Google Scholar

[8]

H. Cho and G. Hwang, Optimal design for dynamic spectrum access in cognitive radio networks under rayleigh fading,, Journal of Industrial and Management Optimization, 8 (2012), 821.  doi: 10.3934/jimo.2012.8.821.  Google Scholar

[9]

E. Chow, L. Hendrix, M. Herberg, S. Itoh, B. Kong, M. Lall and P. Srevens, Pipeline Politics in Asia: The Intersection of Demand, Energy Markets, and Supply Routes,, National Bureau of Asian Research, (2010).   Google Scholar

[10]

Y. Ding and S. Wang, Optimal control of open-channel flow using adjoint sensitivity analysis,, Journal of Hydraulic Engineering-ASCE, 132 (2006), 1215.  doi: 10.1061/(ASCE)0733-9429(2006)132:11(1215).  Google Scholar

[11]

Z. Feng, K. Teo and V. Rehbock, Branch and bound method for sensor scheduling in discrete time,, Journal of Industrial and Management Optimization, 1 (2005), 499.  doi: 10.3934/jimo.2005.1.499.  Google Scholar

[12]

Z. Feng, K. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time,, Automatica, 44 (2008), 1295.  doi: 10.1016/j.automatica.2007.09.024.  Google Scholar

[13]

G. Ferrari-Trecate, A. Buffa and M. Gati, Analysis of coordination in multi-agent systems through partial difference equations,, IEEE Transactions on Automatic Control, 51 (2006), 1058.  doi: 10.1109/TAC.2006.876805.  Google Scholar

[14]

P. Frihauf and M. Krstic, Leader-enabled deployment onto planar curves: A pde-based approach,, IEEE Transactions on Automatic Control, 56 (2011), 1791.  doi: 10.1109/TAC.2010.2092210.  Google Scholar

[15]

R. Glowinski, J. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach,, (Encyclopedia of Mathematics and its Applications) Cambridge University Press, (2008).  doi: 10.1017/CBO9780511721595.  Google Scholar

[16]

H. Hao and P. Barooah, On achieving size-independent stability margin of vehicular lattice formations with distributed control,, IEEE Transactions on Automatic Control, 57 (2012), 2688.  doi: 10.1109/TAC.2012.2191179.  Google Scholar

[17]

H. Hao, P. Barooah and P. Mehta, Stability margin scaling laws for distributed formation control as a function of network structure,, IEEE Transactions on Automatic Control, 56 (2011), 923.  doi: 10.1109/TAC.2010.2103416.  Google Scholar

[18]

J. Kim, K. Kim, V. Natarajan, S. Kelly and J. Bentsman, PdE-based model reference adaptive control of uncertain heterogeneous multiagent networks,, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1152.  doi: 10.1016/j.nahs.2008.09.008.  Google Scholar

[19]

J. Kim, V. Natarajan, S. Kelly and J. Bentsman, Disturbance rejection in robust PdE-based MRAC laws for uncertain heterogeneous multiagent networks under boundary reference,, Nonlinear Analysis: Hybrid Systems, 4 (2010), 484.  doi: 10.1016/j.nahs.2009.11.005.  Google Scholar

[20]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs,, SIAM, (2008).  doi: 10.1137/1.9780898718607.  Google Scholar

[21]

Z. Lin, Distributed Control and Analysis of Coupled Cell Systems,, VDM Verlag, (2008).   Google Scholar

[22]

W. Litvinov, Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions,, Journal of Industrial and Management Optimization, 7 (2011), 291.  doi: 10.3934/jimo.2011.7.291.  Google Scholar

[23]

M. Liu, S. Zang and D. Zhou, Fast leak detection and location of gas pipelines based on an adaptive particle filter,, International Journal of Applied Mathematics and Computer Science, 15 ().   Google Scholar

[24]

M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks (In Applied Mathematics Series),, Princeton University Press, (2010).   Google Scholar

[25]

T. Meurer and M. Krstic, Finite-time multi-agent deployment: A nonlinear pde motion planning approach,, Automatica, 47 (2011), 2534.  doi: 10.1016/j.automatica.2011.08.045.  Google Scholar

[26]

S. Moura and H. Fathy, Optimal boundary control & estimation of diffusion-reaction PDEs,, in Proceeding of the Conference on Decision and Control, (2011), 921.   Google Scholar

[27]

R. Murray, Recent research in cooperative control of multi-vehicle systems,, Journal of Dynamical Systems, (): 571.   Google Scholar

[28]

R. Olfati-Saber and R. Murray, Consensus problems in networks of agents with switching topology and time-delays,, IEEE Transactions on Automatic Control, 49 (2004), 1520.  doi: 10.1109/TAC.2004.834113.  Google Scholar

[29]

P. Parfomak, Pipeline Safety and Security: Federal Programs,, Congress Research Services (CRS) Report for Congress, (2008).   Google Scholar

[30]

M. Rafiee, Q. Wu and A. Bayen, Kalman filter based estimation of flow states in open channels using Lagrangian sensing,, Proceedings of the Conference on Decision and Control, (2009), 8266.  doi: 10.1109/CDC.2009.5400661.  Google Scholar

[31]

W. Ren and Y. Cao, Distributed Coordination of Multi-agent Networks,, (Communications and Control Engineering Series) Springer-Verlag, (2011).   Google Scholar

[32]

A. Sarlette and R. Sepulchre, A PDE viewpoint on basic properties of coordination algorithms with symmetries,, in Proceedings of the Conference on Decision and Control, (2009), 5139.  doi: 10.1109/CDC.2009.5400570.  Google Scholar

[33]

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Edition,, SIAM, (2004).  doi: 10.1137/1.9780898717938.  Google Scholar

[34]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications (Graduate Studies in Mathematics),, American Mathematical Society, (2010).   Google Scholar

[35]

G. Wang and H. Ye, Leakage Detection and Localization of Long Distance Fluid Pipelines,, Tsinghua University Press, (2010).   Google Scholar

[36]

Z. Wang, H. Zhang, J. Feng and S. Lun, Present situation and prospect on leak detection and localization techniques for long distance fluid transport pipeline,, Control and Instruments in Chemical Industry, 30 (2003), 5.   Google Scholar

[37]

S. Woon, V. Rehbock and R. Loxton, Global optimization method for continuous-time sensor scheduling,, Nonlinear Dynamics and Systems Theory, 10 (2010), 175.   Google Scholar

[38]

S. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems,, Optimal Control Applications and Methods, 33 (2012), 576.  doi: 10.1002/oca.1015.  Google Scholar

[39]

K. Yiu, K. Mak and K. Teo, Airfoil design via optimal control theory,, Journal of Industrial and Management Optimization, 1 (2005), 133.  doi: 10.3934/jimo.2005.1.133.  Google Scholar

[40]

C. Yu, B. Li, R. Loxton and K. Teo, Optimal discrete-valued control computation,, Journal of Global Optimization, 56 (2013), 503.  doi: 10.1007/s10898-012-9858-7.  Google Scholar

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